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On Intersecting Conformal Defects

Tom Shachar

Abstract

We study the physics of 2 and 3 mutually intersecting conformal defects forming wedges and corners in general dimension. For 2 defects we derive the beta function of the edge interactions for infinite and semi-infinite wedges and study them in the tricritical model in $d=3-ε$ as an example. We discuss the dependency of the edge anomalous dimension on the intersection angle, connecting to an old issue known in the literature. Additionally, we study trihedral corners formed by 3 planes and compute the corner anomalous dimension, which can be considered as a higher-dimensional analog of the cusp anomalous dimension. We also study 3-line corners related to the three-body potential of point-like impurities.

On Intersecting Conformal Defects

Abstract

We study the physics of 2 and 3 mutually intersecting conformal defects forming wedges and corners in general dimension. For 2 defects we derive the beta function of the edge interactions for infinite and semi-infinite wedges and study them in the tricritical model in as an example. We discuss the dependency of the edge anomalous dimension on the intersection angle, connecting to an old issue known in the literature. Additionally, we study trihedral corners formed by 3 planes and compute the corner anomalous dimension, which can be considered as a higher-dimensional analog of the cusp anomalous dimension. We also study 3-line corners related to the three-body potential of point-like impurities.

Paper Structure

This paper contains 11 sections, 71 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Two Defects: RG on the Edge
  3. Semi-Infinite Plane
  4. Two Intersecting Planes
  5. Wedge Formed by Two Semi-Infinite planes
  6. Example: $\left(\phi^{4}\right)_{\mathcal{D}_1}\cap\left(\phi^{4}\right)_{\mathcal{D}_2}:\left(\phi^{2}\right)_{\mathcal{I}}$ in Tricritical Bulk
  7. Three Defects: Trihedral & 3-Line Corners
  8. Trihedral Corners
  9. The 3-Line Potential
  10. Discussion
  11. List of Integrals

Figures (5)

  • Figure 1: Examples of the defect geometries explored in section \ref{['SEC2']}. The edge $\delta\mathcal{D}$ or the intersection $\mathcal{I}$ are marked with a thick blue line, and $\alpha$ is the intersection angle. For the wedge (middle picture), the intersection is also the common edge of the two planes.
  • Figure 2: The objects studied in section \ref{['SEC3']}. Left: Trihedral Corner. Middle: Extended Trihedral Corner. Right: 3-line Corner. The planes meet at a mutual point, giving rise to the corner anomaly depending upon the three relative angles of the edges, $\alpha_{12},\alpha_{23},\alpha_{13}$.
  • Figure 3: Left: A trihedral corner subtended by the three unit vectors $\hat{e}_{1},\hat{e}_{2},\hat{e}_{3}$. The relative angles $\hat{e}_{a}\cdot\hat{e}_{b}=\cos\left(\alpha_{ab}\right)$ are also drawn. Right: An extended trihedral corner.
  • Figure 4: Left: A 3-line corner defined with $\hat{e}_{1},\hat{e}_{2},\hat{e}_{3}$ and parameterized by $x,y,z \geq0$. Right: An extended 3-line corner, parameterized by $x,y,z \in\mathbb{R}$. Relative angles are defined by $\hat{e}_{a}\cdot\hat{e}_{b}=\cos\left(\alpha_{ab}\right)$.
  • Figure 5: A configuration of 3 line defects with two right angles and $\alpha_{12}=\alpha$ (arbitrarily chosen). Left Panel: $\mathcal{J}_{\frac{\pi}{2}}\left(\alpha\right)$ Plotted for the 3-line corner. Right Panel: $\mathcal{J}_{2\pi}\left(\alpha\right)$ Plotted for the extended 3-line corner.